Chapter 9.3, Problem 7PSA

To determine

To prove: That the product of sides $XY,SV\text{ and }XS,YS$ i.e. $XY·SV=XS·YS$ .

Explanation of Solution

Given information :

A line $YS$ is an altitude of the line $XV.$

  $XV$ is side of $\Delta XVS$ .

  $\angle XSV={90}^{\text{o}}$

Formula used :

Ratio of two corresponding sides of one similar triangle is equal to the ratio of corresponding sides of the second triangle.

Proof :

From $\Delta XYS\text{ and }\Delta SYV$ ,

  $\begin{array}{l}\angle SYV=\angle SYX=90°\\ YS=YS\text{, common side}\\ \angle YVS\text{=}\angle SXY\end{array}$

So, both triangles are similar. Then,

  $\begin{array}{l}\therefore \frac{XY}{XS}=\frac{YS}{SV}\\ \Rightarrow XY·SV=YS·XS\end{array}$

Hence, it has proved.